We have analyzed archival HST F606W images of 18 dwarf elliptical (dE) galaxy candidates in the Coma Cluster. We model the full radial extent of their light-profiles by simultaneously fitting a PSF-convolved Sersic R^(1/n) model and, when necessary, either a central point-source or a central PSF-convolved Gaussian. The luminosities of the central component L_nuc scale with the host galaxy luminosity L_gal such that L_nuc = 10^(4.76 +/- 0.10)*(L_gal/10^7)^(0.87 +/- 0.26). The underlying host galaxies display systematic departures from an exponential model that are correlated with the model-independent host galaxy luminosity and are not due to biasing from the nuclear component. The Pearson correlation coefficient between log(n) and central galaxy surface brightness mu_0 (excluding the flux from extraneous central components) is -0.83 at a significance level of 99.99%. The Pearson correlation coefficient between the logarithm of the Sersic index `n' and the host galaxy magnitude is -0.77 at a significance of 99.9%. We explain the observed relationship between dE galaxy luminosity and inner logarithmic profile slope (gamma-prime) as a by-product of the correlation between luminosity and Sersic index n. Including, from the literature, an additional 232 dE and E galaxies spanning 10 mag in absolute magnitude (M), the dE galaxies are shown to display a continuous sequence with the brighter E galaxies such that mu_0 brightens linearly with M until core formation causes the most luminous (M_B < -20.5 mag) E galaxies to deviate from this relation. The different behavior of dE and E galaxies in the M-mu_e (and M-_e) diagram, and the _e-log(R_e) diagram have nothing to do with core formation, and are in fact expected from the continuous and linear relation between M and mu_0 and M and log(n).